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Harmonic chain far from equilibrium: single-file diffusion, long-range order, and hyperuniformity
by Harukuni Ikeda
Submission summary
| Authors (as registered SciPost users): | Harukuni Ikeda |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2309.03155v3 (pdf) |
| Date submitted: | 2024-07-12 11:42 |
| Submitted by: | Ikeda, Harukuni |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In one dimension, particles can not bypass each other. As a consequence, the mean-squared displacement (MSD) in equilibrium shows sub-diffusion ${\rm MSD}(t)\sim t^{1/2}$, instead of normal diffusion ${\rm MSD}(t)\sim t$. This phenomenon is the so-called single-file diffusion. Here, we investigate how the above equilibrium behaviors are modified far from equilibrium. In particular, we want to uncover what kind of non-equilibrium driving force can suppress diffusion and achieve the long-range crystalline order in one dimension, which is prohibited by the Mermin-Wagner theorem in equilibrium. For that purpose, we investigate the harmonic chain driven by the following four types of driving forces that do not satisfy the detailed balance: (i) temporally correlated noise with the noise spectrum $D(\omega)\sim \omega^{-2\theta}$, (ii) conserving noise, (iii) periodic driving force, and (iv) periodic deformations of particles. For the driving force (i) with $\theta>-1/4$, we observe ${\rm MSD}(t)\sim t^{1/2+2\theta}$ for large $t$. On the other hand, for the driving forces (i) with $\theta<-1/4$ and (ii)-(iv), MSD remains finite. As a consequence, the harmonic chain exhibits the crystalline order even in one dimension. Furthermore, the density fluctuations of the model are highly suppressed in a large scale in the crystal phase. This phenomenon is known as hyperuniformity. We discuss that hyperuniformity of the noise fluctuations themselves is the relevant mechanism to stabilize the long-range crystalline order in one dimension and yield hyperuniformity of the density fluctuations.
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Author comments upon resubmission
Thank you very much for your editorial work, especially for selecting
these four referees. I am happy that the referees have high opinions of
my work. I have carefully studied the referee reports and have revised the paper. I believe that I have fully answered the referees's questions and that they will find the revised
version acceptable for publication.
Sincerely yours,
Harukuni Ikeda
List of changes
Here I list major changes.
1. As suggested by Referees 1 and 4, I clarified the initial conditions in the revised manuscript.
2. As suggested by Referees 1 and 4, I used the center-of-mass frame in the revised manuscript.
3. As suggested by Referee 1, I redefine the discrete Fourier transform so that the waver number to be symmetric.
4. In response to Referee1’s comments, we decided to discuss S(q) only in the solid phase.
5. As suggested by Referee 1, I added a discussion for a free particle driven by the correlated noise in Sec. 3.1.
6. As suggested by Referee 1, I used cosθ instead of 1/secθ in the revised manuscript.
7. As suggested by Referee 1, I added a footnote to explain the inequality in (45).
8. In response to Referee1’s comments, I used O to represent the order parameter instead of R.
9. I modified the introduction as suggested by Referee 1.
10. In response to Referee 2’s comments, in Sec. 7. 6 of the revised manuscript, I discussed that the continuous symmetry breaking in one dimension has not been reported so far, even far from equilibrium.
11. In response to the comment by Referees 3 and 4, we decided to use “conserving noise” to represent the noise introduced in Sec. 4, instead of “center-of-mass conserving noise”.
12. As suggested by Referee 4, we corrected the typos.
Current status:
Reports on this Submission
Report
This paper is much improved from its original version. The author has answered almost all of the criticisms of my previous report. On this basis, I would recommend acceptance, subject to the comments below. (However, I have not read the other reports in detail, so if I have not checked if their criticisms have been answered.)
I do still have a small problem with the setup. Above eq(2) it is stated
"Without loss of generality, we can assume that the equilibrium position of the jth particle is given by $R_j = ja$."
This assumption is not consistent with later statements in the manuscript because the equilibrium position of the jth particle depends on the initialisation. Similarly, it is assumed below eq(7)
"that $\tilde u_q(t)$ reaches a steady state independent from the boundary condition at finite t."
which is not the case.
The point is that if the system is initialised with $x_j=ja$ (or, equivalently, $u_j=0$) then the equilibrium positions are indeed $R_j=j.a$. But if initialised with $x_j=(j+c)a$, (equivalently, u_j=c.a) then the equilibrium positions will be different.
I also do not understand why the authors assume that $\tilde u_q(\pm \infty)= 0$. (I assume here that the $\pm\infty$ refers to the behaviour as a function of time t and not as a function of omega.) Eq(4) is a first-order Langevin equation for $\tilde u_q$ so the behaviour for large positive times is fixed by the initial conditions. Hence it is not appropriate to make this assumption on $\tilde u_q(+\infty)$, which is a random quantity with a non-trivial probability distribution
All this being said, I recommmend the following:
. Initialise the system with $x_j=ja$ [or $u_j=0$] at time $t=-\tau<0$. Since the centre of mass is fixed, this ensures that if the crystal is stable then the equilibrium position R_j of the jth particle is indeed $R_j=ja$.
. Assume that tau is large enough that the system has converged to its steady state at time t==0. (This steady state is now unique because the initial condition was prescribed. Hence this assumption can always be satified.)
. Remove any assumptions on $\tilde u_q(+\infty)$, because these cannot be satisfied in general. Side comment: I don't think that any assumption is needed about "simplifying the Fourier transform", eg note that equation (4) can be solved directly in the time domain to give (for $t>-\tau$):
$$
\tilde u_q(t) = \tilde{u}_q(-\tau) + \int_{-\tau}^t e^{\lambda(s-t)} \tilde\xi(s) ds
$$
From here one can get eq(9,10) without any additional assumptions [except for (7) and the initialisation conditions].
One other small comment:
Just after eq(26), the text reads
"The large-scale fluctuations are highly suppressed. This property is referred to as hyperuniformity [12]."
Near the start of Sec 2.4, it reads
"the density fluctuations are highly suppressed for small q. This property is referred to as hyperuniformity [12]."
It is not needed to repeat the definition of hyperuniformity. (Similar statements appear in other places too, this is not needed.)
Also, the sentence "Recently Galliano et al..." in Section 4.1 is repeated almost verbatim from the introduction (page 2). Such repetition is not needed.
Recommendation
Ask for minor revision